Symplectic Form

graph TD D1["D1 Symplectic form ω\nClosed, nondegenerate 2-form"] D2["D2 Nondegeneracy\nω^n ≠ 0; ω^∧: TM → T*M iso"] D3["D3 Symplectic vector space\n(V, ω) with dim V even"] D4["D4 Symplectomorphism\nφ*ω = ω, preserves form"] T1["T1 Dimension even\ndim M = 2n for symplectic M"] T2["T2 ω^n is volume form\nOriented manifold"] T3["T3 Linear Darboux\n(V,ω) ≅ (R^{2n}, ω_0) standard"] T4["T4 Darboux theorem\nLocally (M,ω) ≅ (R^{2n}, ω_0)"] T5["T5 Symplectic quotient\nM // G inherits symplectic structure"] D1 --> D2 D1 --> D3 D1 --> D4 D2 --> T1 D2 --> T2 D3 --> T3 D1 --> T4 D4 --> T5 T3 --> T4 T1 --> T2 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem